Adding pseudospectral collocation schemes to Moco

[nickbianco]

This PR addes pseudospectral collocation schemes to Moco, as described in #3527.

Brief summary of changes

• Added two new transcription schemes to CasADi, CasOCLegendreGauss and CasOCLegendreGaussRadau for the Gauss Pseudospectral Method (GPM) and Radau Pseudospectral Method (RPM), respectively. The names reflect the Legendre polynomial roots used in each scheme.
• Updates to MocoCasADiSolver to make these schemes available to MocoStudy.

Testing I've completed

• Tests have been added and/or updated in testMocoAnalytic, testMocoActuators, and testMocoInterface to test the new schemes.

Looking for feedback on...

I'm 99% sure I implemented the schemes correctly, however I encountered some difficulty when testing the schemes on the "linear tangent steering" problem in testMocoAnalytic. Initially, running this test with legendre-gauss-3 would cause the problem to almost immediately go into restoration mode in IPOPT. I updated some solver settings, including scaling problem variables, changing the CasADi finite differencing method, and reducing the number of variables (MUMPS was complaining about a memory issue at one point), and this helped with convergence.

My guess is that this problem has some unique numerical difficulties that come up if you're not careful with solver settings. However, I would appreciate a second eye on the testing changes I made so we're fully confident before merging. I did write a Matlab script in CasADi to test the problem formulation, and that seems more stable than the Moco implementation -- but that problem also had the benefit of true automatic differentiation, so it's a bit hard to compare.

I'm assigning @carmichaelong to review this, but @antoinefalisse and @nicos1993, if you have some time to look over the OCP details, that would be super helpful!

CHANGELOG.md (choose one)

• updated.

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This change is 

[antoinefalisse]

Will take a look next week.

[nicos1993]

I will take a look!

[carmichaelong]

Code is looking good overall (I tried to take a careful look at all the details and indices and didn't catch any errors there). There's one (very minor) question and some documentation to address.

Reviewed 5 of 10 files at r1, 1 of 1 files at r2, 1 of 2 files at r3.
Reviewable status: 6 of 11 files reviewed, 2 unresolved discussions (waiting on @nickbianco)

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGauss.cpp line 40 at r2 (raw file):

        const int igrid = imesh * (m_degree + 1);
        // There are no quadrature coefficients at the mesh points (i.e.,
        // quadCoeffs(igrid) = 0.0).

Q: does this imply that the line above DM quadCoeffs(m_numGridPoints, 1); initializes quadCoeffs as all 0's upon construction? (no need to update any comments or code if that's the case)

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGaussRadau.h line 25 at r2 (raw file):

namespace CasOC {

/// Enforce the differential equations in the problem using pseudospectral

these comments seem to be exactly the same as in CasOCLegendreGauss.h

[nickbianco]

Reviewable status: 6 of 11 files reviewed, 2 unresolved discussions (waiting on @carmichaelong and @nickbianco)

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGauss.cpp line 40 at r2 (raw file):

Previously, carmichaelong wrote…

Q: does this imply that the line above DM quadCoeffs(m_numGridPoints, 1); initializes quadCoeffs as all 0's upon construction? (no need to update any comments or code if that's the case)

Yes, that is correct.

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGaussRadau.h line 25 at r2 (raw file):

Previously, carmichaelong wrote…

these comments seem to be exactly the same as in CasOCLegendreGauss.h

I've updated the comments.

[nickbianco]

Thanks @carmichaelong! Ready for review.

Reviewable status: 6 of 11 files reviewed, 2 unresolved discussions (waiting on @carmichaelong and @nickbianco)

[carmichaelong]

Thanks! Just a few minor notes on the documentation.

Reviewed 1 of 10 files at r1, 1 of 2 files at r3, 4 of 4 files at r4.
Reviewable status: all files reviewed (commit messages unreviewed), 3 unresolved discussions (waiting on @nickbianco)

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGauss.h line 41 at r4 (raw file):

/// mesh interval endpoint.
///
/// Control approximation.

Could consider removing the first sentence here, or at least flipping the two sentences so that the details of what's implemented at least come first

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGauss.h line 43 at r4 (raw file):

/// Control approximation.
/// ----------------------
/// In [1, 2], the control is approximated using of basis of Lagrange polynomials

using a basis?

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OpenSim/Moco/MocoCasADiSolver/CasOCLegendreGaussRadau.h line 42 at r4 (raw file):

/// Control approximation.
/// ----------------------
/// In [1, 2], the control is approximated using of basis of Lagrange polynomials

Same comments as in the other header:

• of -> a(?)
• consider putting what's implemented first (either by removing the first sentence or rearranging)

[nickbianco]

@carmichaelong, ready for review (just realized I never pinged you).

[carmichaelong]

Reviewed 2 of 2 files at r5.
Reviewable status: all files reviewed (commit messages unreviewed), 3 unresolved discussions (waiting on @nickbianco)